# DIMACS Focus on Discrete Probability Seminar

## Title:

On a likely shape of the random Ferrers diagram

## Speaker:

- Boris Pittel
- Ohio State University

## Place:

- CoRE Building, Room 431
- Busch Campus, Rutgers University.

## Time:

- 3:30 - 4:30 PM - After the DIMACS Tea
- Thursday, May 8, 1997

**Abstract:**
We study the random partitions of a large integer $n$, under assumption that
all such partitions are equally likely. We use Fristedt's conditioning device
which connects the parts (summands) distribution to the one of a $g$-sequence,
that is a sequence of
independent random variables, each distributed geometrically with a size--
dependent parameter. Confirming a conjecture made by Arratia and Tavar\'e, we
prove that the joint distribution of counts of parts with size
at most $s_n \ll n^{1/2}$
(at least $s_n\gg n^{1/2}$, resp.) is close---in terms of the total variation
distance---to the distribution of the first $s_n$ components of the $g$-
sequence (of the $g$ sequence minus the first $s_n-1$ components, resp.). We
supplement these results with the estimates for the middle--size parts
distribution, using the analytical tools revolving around Hardy--Ramanujan
formula for the partition function. Taken together, the estimates lead to an
asymptotic description of the random Ferrers diagram, close to the one obtained
earlier by Szalay and Tur\'an. As an application, we simplify considerably
and strengthen
Szalay--Tur\'an's formula for the likely degree of an irreducible
representation of the symmetric group $S_n$. We show further that both the size
of a random conjugacy class and the size of the centraliser for every element
>from the class are double exponentially distributed in the limit. We prove that
a continuous time process that describes the random fluctuations of the
diagram boundary from the deterministic approximation converges to a Gaussian
(non--Markov) process with continuous sample path. Convergence is such that
it implies weak convergence of every integral functional from a broad class.
To demonstrate applicability of this general result, we prove that the
eigenvalue distribution for Diaconis--Shahshahani's card-shuffling Markov chain
is asymptotically Gaussian with zero mean, and variance of order $n^{-3/2}$.

Document last modified on May 5, 1997